Could anybody help me with the following question ?

Assume we are given:

(1) a finite order (linear) automorphism $g$ of the projective space $\mathbb{P}^r$,

(2) a closed algebraic subvariety $Z \subset \mathbb{P}^r$ of codimension at least 2.

Is it always possible to find a line in $\mathbb{P}^r$ which is stable under $g$ and does not meet $Z$?

Thanks in advance!

Could anybody help me with the following question ?

Assume we are given:

(1) a finite order (linear) automorphism $g$ of the complex projective space $\mathbb{P}^r$,

(2) a closed algebraic subvariety $Z \subset \mathbb{P}^r$ of codimension at least 2.

The problem is to find lines in $\mathbb{P}^r$ which are stable under $g$ and do not meet $Z$. This is not always possible. However, one can embed $\mathbb{P}^r$ in a bigger dimension projective space $\mathbb{P}^N$ by the Veronese embedding of degree $d$ and extend $g$ to a finite order automorphism of $\mathbb{P}^N$. Is it true, for sufficiently large $d$, that there exists a $g$-invariant line in $\mathbb{P}^N$ which does not meet the image of $Z$?

Thanks in advance!